状态空间气动弹性模型及其在颤振计算中的应用

本文评述了已有的状态空间气动弹性模型,并对它们提出了一些改进意见。本文还提出了一种新的状态空间气动弹性模型。用两种机翼为例进行了颤振计算。算例表明:这种新的状态空间气动弹性模型是一种精度较高而阶数较低的模型。本文还提出了用状态空间气动弹性模型进行颤振计算时自动识别模态及自动确定颤振点的方法。

STATE-SPACE AEROELASTIC MODELING AND ITS APPLICATION IN FLUTTER CALCULATION

Three rational approximations of unsteady loads （Roger's, matrix Pade's and Karpel's） are reviewed for designing the control law of an active flutter suppression system. Three indices for evaluating the fitting accuracy are proposed.It is shown that the accuracy of matrix Pade approximation is unsatisfactory even though its order is much lower than Roger's. So does the order of Karpel's, furthermore its fitting accuracy precedes that of matrix Pade's, but improvement of the accuracy is limited by nonlinearity of the fitting equation. The accuracy of Roger approximation is the best because of independent determination of elements in the same matrix, but its order is also the highest.An improved Karpel's method is proposed, which reduces a complicated nonlinear fitting problem to a linear one（Eq. 11） by means of matrix transformation. Numerical results show that the fitting accuracy is enhanced considerably and calculation is simplified.A new approximation（Eq. 14）is provided in a form similar to Roger's in order to lessen the correlation of the same matrix as far as possible. In addition, an optimal procedure for determining the constants involved in the denominators of each fraction is adopted. Both of them improve the fitting accuracy. However, each fraction in this new approximation only corresponds to one augmented state variable instead of n augmented state variables in Roger's （n is the order of flutter equations of motion）, hence the order of the model is decreased considerably.The results of two numerical examples（Tables 1 and 2 ）show that the accuracy of new approximation is comparable to Roger's, but its order is only about half of Roger's.The first and second derivatives of eigenvalues of flutter determinant in state space with respect to velocity are derived （Eqs. 17 and 18）, whereby a method of auto-identifying modals and auto-determining flutter point is presented. Numerical results of a flutter calculation （Fig. 1a and 1b） show that this method can automatically identify modals even in rather complicated situations.